Is it possible to render and composite certain materials before others?
Yes and no.

You can not render certain materials before other ones, both cycles and the blender internal will render your whole scene in one go. You can't say render my cube before the ball.

However you can use render layers. Render layers brake up your scene in to pieces that get rendered separately. Then after it is done rendering everything you can combine, and manipulate them way you want in the compositor.

Here I have a simple scene, a cube, a ball and a background. The important thing is that each object is on its own render layer, the cube, the ball and the background, are all on their own render layers.



Those three images (click for full size) are what you get if you render them using separate render layers. Note on the background layer it renders the part of the background that would normally be hidden but the ball and cube, yet the shadows stay.

Now in the compositor I combined these using alpha over nodes, to get the final rendered image. The benefit here is that you can apply any effects to just one piece. For example, I changed the color of the cube.





Your other option, which will work just as well in a simple case like my example is to use an object or material index pass. This works a bit differently then separating the objects out on different render layers, here they all are rendered in one pass, but the addition of the Object Index pass works like a mask for the objects.
Do note that you first must set the pass on each object. In the Object tab of the properties window, under the Relations section you will find the Pass Index value.


After all the passes are set correctly for the objects, render and let the fun in the compositor begin.

Here are the nodes to use a Object Index pass as a mask, which will isolate the specific object.

It may look complicated but, it is really just the Object Index pass as the factor in the Mix node. The important part is the ID Mask node. The number you put in there corresponds to the number you set the object pass to.
All that I'm doing is some color, and blurring manipulations on the original image. Then Using the ID Mask to pull the correct mask out of the Object Index pass. Which gets feed in to the mix nodes, to determine what part of the image gets manipulated.